Polytope of Type {18,6} with group SmallGroup(648,297)

Overview

Group: SmallGroup(648,297)
Rank: 3
Schläfli Type: {18,6}
Vertices, edges, …: 54, 162, 18
Order of s₀s₁s₂: 18
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

2-fold

{18,6}*324a

3-fold

6-fold

{6,6}*108

9-fold

18-fold

{9,2}*36

27-fold

54-fold

81-fold

{2,2}*8

Covers minimal covers in bold

2-fold

3-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^3*s0*s2*(s1*s0)^2*s2*s1> of order 3

6 facets

6 of {18}*36

18 vertex figures

18 of {6}*12

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3

6 facets

6 of {18}*36

30 vertex figures

12 of {6}*12
18 of {2}*4

Representations

Permutation Representation (GAP)

s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 61)( 29, 63)( 30, 62)( 31, 58)( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 70)( 38, 72)( 39, 71)( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 79)( 47, 81)( 48, 80)( 49, 76)( 50, 78)( 51, 77)( 52, 73)( 53, 75)( 54, 74)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(109,142)(110,144)(111,143)(112,139)(113,141)(114,140)(115,136)(116,138)(117,137)(118,151)(119,153)(120,152)(121,148)(122,150)(123,149)(124,145)(125,147)(126,146)(127,160)(128,162)(129,161)(130,157)(131,159)(132,158)(133,154)(134,156)(135,155);;
s1 := (  1, 28)(  2, 29)(  3, 30)(  4, 34)(  5, 35)(  6, 36)(  7, 31)(  8, 32)(  9, 33)( 10, 48)( 11, 46)( 12, 47)( 13, 54)( 14, 52)( 15, 53)( 16, 51)( 17, 49)( 18, 50)( 19, 38)( 20, 39)( 21, 37)( 22, 44)( 23, 45)( 24, 43)( 25, 41)( 26, 42)( 27, 40)( 55, 61)( 56, 62)( 57, 63)( 64, 81)( 65, 79)( 66, 80)( 67, 78)( 68, 76)( 69, 77)( 70, 75)( 71, 73)( 72, 74)( 82,109)( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,129)( 92,127)( 93,128)( 94,135)( 95,133)( 96,134)( 97,132)( 98,130)( 99,131)(100,119)(101,120)(102,118)(103,125)(104,126)(105,124)(106,122)(107,123)(108,121)(136,142)(137,143)(138,144)(145,162)(146,160)(147,161)(148,159)(149,157)(150,158)(151,156)(152,154)(153,155);;
s2 := (  1, 91)(  2, 93)(  3, 92)(  4, 94)(  5, 96)(  6, 95)(  7, 97)(  8, 99)(  9, 98)( 10, 82)( 11, 84)( 12, 83)( 13, 85)( 14, 87)( 15, 86)( 16, 88)( 17, 90)( 18, 89)( 19,100)( 20,102)( 21,101)( 22,103)( 23,105)( 24,104)( 25,106)( 26,108)( 27,107)( 28,118)( 29,120)( 30,119)( 31,121)( 32,123)( 33,122)( 34,124)( 35,126)( 36,125)( 37,109)( 38,111)( 39,110)( 40,112)( 41,114)( 42,113)( 43,115)( 44,117)( 45,116)( 46,127)( 47,129)( 48,128)( 49,130)( 50,132)( 51,131)( 52,133)( 53,135)( 54,134)( 55,145)( 56,147)( 57,146)( 58,148)( 59,150)( 60,149)( 61,151)( 62,153)( 63,152)( 64,136)( 65,138)( 66,137)( 67,139)( 68,141)( 69,140)( 70,142)( 71,144)( 72,143)( 73,154)( 74,156)( 75,155)( 76,157)( 77,159)( 78,158)( 79,160)( 80,162)( 81,161);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(162)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 61)( 29, 63)( 30, 62)( 31, 58)( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 70)( 38, 72)( 39, 71)( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 79)( 47, 81)( 48, 80)( 49, 76)( 50, 78)( 51, 77)( 52, 73)( 53, 75)( 54, 74)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(109,142)(110,144)(111,143)(112,139)(113,141)(114,140)(115,136)(116,138)(117,137)(118,151)(119,153)(120,152)(121,148)(122,150)(123,149)(124,145)(125,147)(126,146)(127,160)(128,162)(129,161)(130,157)(131,159)(132,158)(133,154)(134,156)(135,155);
s1 := Sym(162)!(  1, 28)(  2, 29)(  3, 30)(  4, 34)(  5, 35)(  6, 36)(  7, 31)(  8, 32)(  9, 33)( 10, 48)( 11, 46)( 12, 47)( 13, 54)( 14, 52)( 15, 53)( 16, 51)( 17, 49)( 18, 50)( 19, 38)( 20, 39)( 21, 37)( 22, 44)( 23, 45)( 24, 43)( 25, 41)( 26, 42)( 27, 40)( 55, 61)( 56, 62)( 57, 63)( 64, 81)( 65, 79)( 66, 80)( 67, 78)( 68, 76)( 69, 77)( 70, 75)( 71, 73)( 72, 74)( 82,109)( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)( 91,129)( 92,127)( 93,128)( 94,135)( 95,133)( 96,134)( 97,132)( 98,130)( 99,131)(100,119)(101,120)(102,118)(103,125)(104,126)(105,124)(106,122)(107,123)(108,121)(136,142)(137,143)(138,144)(145,162)(146,160)(147,161)(148,159)(149,157)(150,158)(151,156)(152,154)(153,155);
s2 := Sym(162)!(  1, 91)(  2, 93)(  3, 92)(  4, 94)(  5, 96)(  6, 95)(  7, 97)(  8, 99)(  9, 98)( 10, 82)( 11, 84)( 12, 83)( 13, 85)( 14, 87)( 15, 86)( 16, 88)( 17, 90)( 18, 89)( 19,100)( 20,102)( 21,101)( 22,103)( 23,105)( 24,104)( 25,106)( 26,108)( 27,107)( 28,118)( 29,120)( 30,119)( 31,121)( 32,123)( 33,122)( 34,124)( 35,126)( 36,125)( 37,109)( 38,111)( 39,110)( 40,112)( 41,114)( 42,113)( 43,115)( 44,117)( 45,116)( 46,127)( 47,129)( 48,128)( 49,130)( 50,132)( 51,131)( 52,133)( 53,135)( 54,134)( 55,145)( 56,147)( 57,146)( 58,148)( 59,150)( 60,149)( 61,151)( 62,153)( 63,152)( 64,136)( 65,138)( 66,137)( 67,139)( 68,141)( 69,140)( 70,142)( 71,144)( 72,143)( 73,154)( 74,156)( 75,155)( 76,157)( 77,159)( 78,158)( 79,160)( 80,162)( 81,161);
poly := sub<Sym(162)|s0,s1,s2>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References

None.

to this polytope.